No. The closest thing in J.R.R Tolkiens legendarium to a unicorn was probably the maeras who were horses that was “handmade” by the valar orôme, the valar of hunting and the animals of nature. read more
The maera does not have a horn on the top of his head but it always has the sleek white hide that the unicorn almost always has plus they were made by a divine being which makes them more stronger and more intelligent than the normal horse. read more
The "mythology", as all mythologies do, supposes that everything is real. Hence, to have a unicorn in Middle-earth, you have to have a place for it. read more